Question

A 20 kg object is acted on by a conservative force given by f = −3.0x − 5.0x2, with f in newtons and x in meters. take the potential energy associated with the force to be zero when the object is at x = 0.

Answer 1

Given that the conservative force is given by
`F=-3.0x-5.0x^2`, the potential energy, U, is given by:


`F= -(dU)/(dx) \\ \\ \Rightarrow dU=-Fdx \\ \\ \Rightarrow U= -\int\limits^b_a {F} \, dx`

Given that the potential energy associated with the force is zero when the object is at x = 0, the potential energy of the system associated with the force when the object is at x = 5.0 m is given by:


`U=- \int\limits^5_0 {\left(-3.0x-5.0x^2\right)} \, dx \\ \\ =-\left[-1.5x^2- (5)/(3) x^3\right]^5_0=-\left(-1.5(5)^2- (5)/(3) (5)^3\right) \\ \\ =-(-1.5(25)- (5)/(3) (125))=-(-37.5-208.33) \\ \\ =-(-245.83)=\bold{245.8J}`